Galilean Transformations

Introduction

• Galilean Transformation: Mathematical method for transforming quantities between two inertial reference frames.
• Use case: Transforming the position of an object between a stationary frame and a moving frame with velocity ( V ) along the x-axis.

Position Transformation

• Frame Coordinates:
  • Frame 2: ( x' , y' , z' )
  • Frame 1: ( x , y , z )
• Transformation Equations:
  • ( x = x' + Vt )
  • Initial assumption: Frames coincided at time ( t = 0 ).
  • Movement: Frame 2 moves along the x-axis with velocity ( V ), no movement along y and z axes.

Velocity Transformation

• Purpose: To find the velocity of an object in different reference frames.
• Diagram Overview:
  • At ( t = 0 ): Frame 1 and Frame 2 coincide (origins overlap).
  • At ( t = t_2 ): Frame 1 is stationary, Frame 2 moves right along x-axis with velocity ( V ).
• Particle P: Point P with velocity vector ( \mathbf{W'} ) in Frame 2.
• Velocity Components in Frame 2:
  • ( W'_x = \frac{d x'}{d t} )
  • ( W'_y = \frac{d y'}{d t} )
  • ( W'_z = \frac{d z'}{d t} )

Using Position Transformations for Velocity

• Objective: Determine velocity vector in Frame 1 (( \mathbf{W} )).
• Process:
  • Velocity in x-axis for Frame 1: ( W_x = \frac{d x}{d t} ).
  • Use position transformation: ( x = x' + Vt ).
    • Substitute: ( W_x = \frac{d (x' + Vt)}{d t} ).
    • Result: ( W_x = \frac{d x'}{d t} + V ).
  • For y and z axes:
    • ( W_y = \frac{d y}{d t} = \frac{d y'}{d t} ).
    • ( W_z = \frac{d z}{d t} = \frac{d z'}{d t} ).

Galilean Velocity Transformation Equations

• Equations:
  • ( W_x = W'_x + V ).
  • ( W_y = W'_y ).
  • ( W_z = W'_z ).
• Summary: These equations provide the velocity transformation between two reference frames using Galilean transformations.